Subsections

• Exercises

Implicit Differentiation

The implicitdiff command can be used to find derivatives of implicitly defined functions. The syntax is as follows

  > implicitdiff(f,y,x);

where f is an expression or equation, y is the dependent variable and x is the independent variable. Thus the command as just stated would compute $$\frac{dy}{dx}$$. If f is given as an expression Maple will assume that the implicit equation is f = 0. Check the results of the following commands.

  > f:=x^2*y^2+y^3;

  > implicitdiff(f,y,x);

  > g:=x^2+y^3=1;

  > implicitdiff(g,y,x);

Second derivatives can also be taken with implicitdiff. The following command computes $$\frac{d^2y}{dx^2}$$.

  > implicitdiff(g,y,x,x);

Maple also has a command for plotting implicitly defined functions. It is in the package plots which must be called before using the command.

  > with(plots);

  > implicitplot(x^2-y^2=1,x=-3..3,y=3..3);

Exercises

1.

The graph of the equation x²-xy+y²=9 is a rotated ellipse.

a.

Plot the graph of the curve.

b.

Using implicit differentiation find y'.

c.

Show that the tangents to this curve at the two points where it intersects the x-axis are parallel.

d.

Is that true for the tangents to this ellipse at any pair of points obtained by intersecting the ellipse with a line passing through the origin? Justify your answer.

2.

Consider the following implicitly defined functions y. Find the equations of the line tangents at the given points

$$\cos^3x -\cos^3y=\sin (x+y), \;\;\; (0,0)$$

$$x^5+y^5=5x^2y^2, \;\;\; (-5,5)$$

$$y^2=x^2(x+7), \;\;\; (-3,6)$$

3.

A lighthouse is one mile out to the sea from a stretch of straight shore. Its beacon rotates three times per minute. Let A be the point on the shore nearest the lighthouse and B a point on the shore three miles far from A. How fast does the spot of light from the beacon travel (along the shore AB) as it passes point B?

DUE DATE: FEBRUARY 15 (in class).